In mathematics, the Artin–Rees lemma is a basic result about modules over a Noetherian ring, along with results such as the Hilbert basis theorem. It was proved in the 1950s in independent works by the mathematicians Emil Artin and David Rees; a special case was known to Oscar Zariski prior to their work. An intuitive characterization of the lemma involves the notion that a submodule N of a module M over some ring A with specified ideal I holds a priori two topologies: one induced by the topology on M, and the other when considered with the I-adic topology over A. Then Artin-Rees dictates that these topologies actually coincide, at least when A is Noetherian and M finitely-generated. One consequence of the lemma is the Krull intersection theorem. The result is also used to prove the exactness property of completion. The lemma also plays a key role in the study of ℓ-adic sheaves.

Statement

Let I be an ideal in a Noetherian ring R; let M be a finitely generated R-module and let N a submodule of M. Then there exists an integer k ≥ 1 so that, for n ≥ k,

Proof

The lemma immediately follows from the fact that R is Noetherian once necessary notions and notations are set up. For any ring R and an ideal I in R, we set (B for blow-up.) We say a decreasing sequence of submodules is an I-filtration if ; moreover, it is stable if for sufficiently large n. If M is given an I-filtration, we set ; it is a graded module over B_I R. Now, let M be a R-module with the I-filtration M_i by finitely generated R-modules. We make an observation Indeed, if the filtration is I-stable, then B_I M is generated by the first k+1 terms and those terms are finitely generated; thus, B_I M is finitely generated. Conversely, if it is finitely generated, say, by some homogeneous elements in, then, for n \ge k, each f in M_n can be written as with the generators g_{j} in. That is,. We can now prove the lemma, assuming R is Noetherian. Let M_n = I^n M. Then M_n are an I-stable filtration. Thus, by the observation, B_I M is finitely generated over B_I R. But is a Noetherian ring since R is. (The ring R[It] is called the Rees algebra.) Thus, B_I M is a Noetherian module and any submodule is finitely generated over B_I R; in particular, B_I N is finitely generated when N is given the induced filtration; i.e.,. Then the induced filtration is I-stable again by the observation.

Krull's intersection theorem

Besides the use in completion of a ring, a typical application of the lemma is the proof of the Krull's intersection theorem, which says: for a proper ideal I in a commutative Noetherian ring that is either a local ring or an integral domain. By the lemma applied to the intersection N, we find k such that for n \ge k, Taking n = k+1, this means or N = IN. Thus, if A is local, N = 0 by Nakayama's lemma. If A is an integral domain, then one uses the determinant trick (that is a variant of the Cayley–Hamilton theorem and yields Nakayama's lemma): In the setup here, take u to be the identity operator on N; that will yield a nonzero element x in A such that x N = 0, which implies N = 0, as x is a nonzerodivisor. For both a local ring and an integral domain, the "Noetherian" cannot be dropped from the assumption: for the local ring case, see local ring. For the integral domain case, take A to be the ring of algebraic integers (i.e., the integral closure of \mathbb{Z} in \mathbb{C}). If \mathfrak p is a prime ideal of A, then we have: for every integer n > 0. Indeed, if, then for some complex number \alpha. Now, \alpha is integral over \mathbb{Z}; thus in A and then in, proving the claim.